A reverse osmosis unit (R.O.) is being used to concentrate lactose from cottage

Question

A reverse osmosis unit (R.O.) is being used to concentrate lactose from cottage cheese waste which is flowing at 1000 pounds per hour. The waste stream contains 5% lactose. The outputs from the unit are a concentrate stream with 25% lactose and a water stream that contains a residual 0.1% lactose. Part of the concentrate stream coming from the R.O. is recycles back to the input to prevent clogging. This recycle stream after mixing with the 5% input stream has a net lactose concentration of 12% before entering the R.O. (a) How many pounds per hour of 25% concentrate is produced? (b) How many pounds per hour of lactose are "lost" in the water stream? (c) How many pounds per hour of 25% concentrate is recycled through the R.O.?

Explanation / Answer

Let P= Product and W= Waste, F= Feed

Writing overall lactose balance, 1000*5/100 = P*25/100+W*0.1/100, 5000 = P*25+W*0.1   (1)  

Writing overall balance, 1000 = P+W , P= 1000-W (2)

Substituting Eq.2 in Eq.1 gives 5000= (1000-W)*25+W*0.1 = 5000 = 25000- 25W+W*0.1

W*(25-0.1)= 25000-5000, W = 20000/24.9=803 lb/hr, P= 1000-803= 197 lb/hr

Concentrate produced, P= 803 lb/hr, lactase last = 197*0.10/100 = 0.197 lb/hr

Let R= Recycle

Writing balance of lactose across the membrane R*25/100 + 1000*5/100 = (R+1000)*12/100

Hence R*25+5000=12*(R+1000), R*25+5000= 12R+12000, 13R= 7000, R= 7000/13= 538.4 lb/hr

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